Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{y^2 - 7y - 18}{y^2 - 2y - 8}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{y^2 - 7y - 18}{y^2 - 2y - 8} = \dfrac{(y - 9)(y + 2)}{(y - 4)(y + 2)} $ Notice that the term $(y + 2)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(y + 2)$ gives: $z = \dfrac{y - 9}{y - 4}$ Since we divided by $(y + 2)$, $y \neq -2$. $z = \dfrac{y - 9}{y - 4}; \space y \neq -2$